log5(3/5)+3log(15/2)_log(81/8)

Assuming all the logs are base 5, we have

(log3 - log5) + 3(log3 + log5 - log2) - (4log3 - 3log2)
2log5 = 2

To simplify the given expression, let's break it down step by step:

Step 1: Simplify log5(3/5)
Using the logarithmic property logb(x/y) = logb(x) - logb(y), we can rewrite log5(3/5) as log5(3) - log5(5).
Since log5(5) = 1 (any number raised to the power of 1 is equal to itself), we can simplify this to log5(3) - 1.

Step 2: Simplify 3log(15/2)
Using the logarithmic property logb(x^n) = n * logb(x), we can rewrite 3log(15/2) as log(15/2)^3.
Since log(15/2)^3 is the logarithm of a single number, we can evaluate the exponent (cube of the numerator) inside the logarithm: log(15^3/2^3).
This can be further simplified as log(3375/8).

Step 3: Simplify log(81/8)
Using the logarithmic property logb(x/y) = logb(x) - logb(y), we can rewrite log(81/8) as log(81) - log(8).
Since both 81 and 8 can be written as powers of 3, we have log(3^4) - log(2^3).
Using the logarithmic property logb(x^n) = n * logb(x), this simplifies to 4log(3) - 3log(2).

Putting it all together, the expression becomes:

(log5(3) - 1) + log(3375/8) - (4log(3) - 3log(2))

Now, we can simplify further:

Removing the parentheses, the expression becomes:

log5(3) - 1 + log(3375/8) - 4log(3) + 3log(2)

Combining like terms:

log5(3) - 4log(3) + 3log(2) + log(3375/8) - 1

Finally, using the logarithmic property logb(x * y) = logb(x) + logb(y), we can rewrite this as:

log5(3) - 4log(3) + 3log(2) + log(3375) - log(8) - 1

This is the simplified form of the given expression.

To simplify the expression log5(3/5) + 3log(15/2) - log(81/8), we will use the properties of logarithms.

1. Let's start by simplifying log5(3/5). This can be rewritten as log5(3) - log5(5).

2. Next, we simplify 3log(15/2). Using the power rule of logarithms, we can rewrite this as log(15/2)^3.

3. Continuing to simplify, log(15/2)^3 can be further written as log(15^3/2^3).

4. Finally, we simplify log(81/8) as log81 - log8.

Now, let's put everything together:

log5(3/5) + 3log(15/2) - log(81/8)
= log5(3) - log5(5) + log(15^3/2^3) - log81 + log8

We can simplify each of these logarithmic expressions further using the properties of logarithms if necessary.